Question

Find the triangles with angles `A, B, C` and correspondingly opposite sides `a,b.c` such that `(aA+bB+cC)/(a+b+c)` has a minimum. Does this expression have a maximum?

Answer 1

The minimum would be zero. The maximum is infinity.

Explanation:

As given in the problem, the sides could shrink to a point, regardless of the angles. Similarly, there is no bound on the length of any of the sides.
For example, an equilateral triangle could be expanded infinitely. The ratio would continue to increase.

Answer 2

See below.

Explanation:

Considering the Chevishef's inequality

`(1/n sum_(k=1)^n x_k) (1/n sum_(j=1)^n y_j) le 1/n sum_(k=1)^n x_k y_k`

for

`0 lt x_1 le x_2 le cdots le x_n`
`0 lt y_1 le y_2 le cdots le y_n`

we can arrange

`a le b le c` and correspondingly
`A le B le C`

so we have

`1/3(a+b+c)(A+B+C) le (aA+bB+cC)`

and then

`(aA+bB+cC)/(a+b+c) ge 1/3((a+b+c)(A+B+C))/(a+b+c) = 1/3(A+B+C) = pi/3`

The maximum determination (not attainable) is left as an exercise.